Problem: Simplify and expand the following expression: $ \dfrac{2q - 1}{2q + 4}+\dfrac{5q}{q - 5} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2q + 4)(q - 5)$ Multiply the first term by $\dfrac{q - 5}{q - 5}$ $ \begin{align*} \dfrac{2q - 1}{2q + 4} \times \dfrac{q - 5}{q - 5} & = \dfrac{(2q - 1)(q - 5)}{(2q + 4)(q - 5)} \\ & = \dfrac{2q^2 - 11q + 5}{(2q + 4)(q - 5)}\end{align*} $ Multiply the second term by $\dfrac{2q + 4}{2q + 4}$ $ \begin{align*} \dfrac{5q}{q - 5} \times \dfrac{2q + 4}{2q + 4} & = \dfrac{(5q)(2q + 4)}{(q - 5)(2q + 4)} \\ & = \dfrac{10q^2 + 20q}{(q - 5)(2q + 4)}\end{align*} $ Now we have: $ = \dfrac{2q^2 - 11q + 5}{(2q + 4)(q - 5)} + \dfrac{10q^2 + 20q}{(q - 5)(2q + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2q^2 - 11q + 5 + 10q^2 + 20q}{(2q + 4)(q - 5)} $ $ = \dfrac{12q^2 + 9q + 5}{(2q + 4)(q - 5)}$ Expand the denominator: $ = \dfrac{12q^2 + 9q + 5}{2q^2 - 6q - 20}$